Essential norms of composition operators between Bloch type spaces

For $\alpha>0$, the $\alpha$-Bloch space is the space of all analytic functions $f$ on the unit disk $D$ satisfying

$$\Vert f\Vert _{B^{\alpha}}=\sup_{z\in D}\vert f'(z)\vert(1-\vert z\vert^2)^{\alpha}<\infty.$$

Let $\varphi$ be an analytic self-map of $D$. We show that for $0<\alpha,\beta<\infty$, the essential norm of the composition operator $C_{\varphi}$ mapping from $B^{\alpha}$ to $B^{\beta}$ can be given by the following formula:

$$\Vert C_{\varphi}\Vert _e=\left(\frac{e}{2\alpha}\right)^{\alpha}\limsup_{n\to\infty} n^{\alpha-1}\Vert\varphi^n\Vert _{B^{\beta}}.$$